Use Green's Theorem to evaluate the integral [(y-x) dx + (2x−y)dy where C is the boundary of the region C lying inside the rectangle bounded by x = -8,x = 8,y=-4, y = 4, and outside the square bounded by x = -1,x= = 1, y = -1, and y = 1.

Q: 2. [L-4) 5 (y+3)*

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Q: valuate the X = (3-9) JJ²

A: We solve by using double integral.

Q: hat is the area of the region bounded by the line x = 4 and the curve x³ = x² + 2xy - y² = 0. ht:…

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Q: 14. Evaluate the integral: (x² + 3x)° dx A. (x* + 30)' + C 3 x* + 3x' + C C. +C (x² + 3x) + C В. D.

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Q: 1200 1100 1000 900 800 700 600 500 400 300 200 100 100 Find the (exact) area under the curve y = 4x*…

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Q: Evaluate ([7y'e*+ dA where D is the region bounded by y = 2 x , x=9 and the x-axis.

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Q: 5. Find the area between the curve y, =x² - 4x +12 and y, = x² from 0<x<4

A: Given curves are y1=x2-4x+12y2=x2 Also 0≤x≤4

Q: Evaluate the following integrals: dx

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Q: I. Integral Evaluation. (40 points) Evaluate the following five integrals. | (5z* - 8z + 6z) dr. 1.…

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Q: 6. Fill in the blank: The length of the graph of y = x² on [2, 4] is equal to the area under the…

A: y=x2 on the interval 2, 4. Differentiating with respect to x, we get dydx=2x Now length of the curve…

Q: 3) Find the area of the surface formed by revolving the graph f(x) = ;x³ about the x-axis over the…

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Q: 5. Evaluate the integral of 1 y¹/³(x³ + 1) over the region which is bounded by the x axis, the line…

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Q: Which integral corresponds to the area of the shaded region? The parabola is y = x? –1 while the…

A: Given query is to find the the integral for the area of shaded region.

Q: 1. Find the total area bounded by the curves y=x²-3x and y = x³ + x² −12x.

A: Given curves are y=x2-3x y=x3+x2-12x Take  y=fxHere,   fx=x2-3x y=gxHere,   gx= x3+x2-12x

Q: By using u = In x, evaluate the integrals In x dx.

A: ∫x-1xlnx dx Now put, u=lnx⇒dudx=1x⇒du=1x dx